Now suppose that \(\rho <\rho _S(\delta )\) . Since \(G(\delta _d,\rho _d) \rightarrow G(\delta ,\rho ) <0\) , there is a constant \(c\)  such that \(G(\delta _d,\rho _d) \le c<0\)  for all large \(d\) . Hence, (REF ) implies that \(\lim _{d\rightarrow \infty } \left[\binom{N}{k} -{\mathbb {E}}\,f_k(D_N)\right] =0\) . This was proved in a different way in [1]}.
